Deconstructing Optimal f: The Mathematical Core of Geometric Growth
The Black Book of Day Trading Strategies
1,000 complete strategies · 31 chapters · Full trade plans
Introduction
While the conceptual appeal of Optimal f is undeniable, its practical implementation and appreciation require a deeper understanding of its mathematical underpinnings. At its core, Optimal f is an optimization problem that seeks to maximize the geometric mean of returns, a concept that is fundamentally linked to the long-term compounding of capital. This article deconstructs the mathematical framework of Optimal f, exploring the nuances of the Terminal Wealth Relative (TWR), the role of the Holding Period Return (HPR), and the iterative process of identifying the optimal trading fraction. A thorough grasp of these mathematical concepts is not merely an academic exercise; it is a prerequisite for the intelligent application and adaptation of this effective position-sizing methodology.
The Primacy of Geometric Mean in Portfolio Growth
To understand Optimal f, one must first appreciate the distinction between arithmetic and geometric means in the context of investment returns. The arithmetic mean, while simple to calculate, can be misleading as it does not account for the effects of volatility and compounding. The geometric mean, on the other hand, represents the constant rate of return that, if compounded over a period, would yield the same terminal wealth as the actual sequence of returns. It is, therefore, a more accurate measure of the true growth rate of an investment.
The formula for the geometric mean of a series of returns is:
Geometric Mean = ( (1 + R1) * (1 + R2) * ... * (1 + Rn) )^(1/n) - 1
Geometric Mean = ( (1 + R1) * (1 + R2) * ... * (1 + Rn) )^(1/n) - 1
Where:
- R(i) is the return for the ith period.
- n is the number of periods.
Optimal f is a direct application of this principle. It seeks to find the fixed fraction of capital (f) that, when applied to a sequence of trades, maximizes the geometric mean of the resulting portfolio returns. This is achieved by maximizing the Terminal Wealth Relative (TWR), which is mathematically equivalent to maximizing the geometric mean.
The Holding Period Return (HPR) and the TWR Equation
The Holding Period Return (HPR) is the return on a single trade, expressed as a multiple of the initial investment. For a given trade, the HPR is calculated as:
HPR = 1 + (f * (-Trade / BiggestLoss))
HPR = 1 + (f * (-Trade / BiggestLoss))
Where:
- f is the fraction of capital being risked.
- Trade is the profit or loss of the trade.
- BiggestLoss is the largest single loss in the trade sequence.
The TWR is the product of the HPRs for all trades in the sequence:
TWR = HPR1 * HPR2 * ... * HPRn
TWR = HPR1 * HPR2 * ... * HPRn
The objective of Optimal f is to find the value of f that maximizes this TWR. This is typically done through an iterative process, where different values of f (from 0 to 1) are tested, and the corresponding TWR is calculated. The value of f that yields the highest TWR is the Optimal f.
A Practical Example: Iterating to Find Optimal f
Let's revisit the trade sequence from the previous article:
| Trade | P/L ($) |
|---|---|
| 1 | +5,000 |
| 2 | -2,000 |
| 3 | +8,000 |
| 4 | -3,000 |
| 5 | +12,000 |
| 6 | -4,000 |
| 7 | +6,000 |
| 8 | -1,000 |
| 9 | +10,000 |
| 10 | -5,000 |
The biggest loss is $5,000. We can now create a table to calculate the TWR for different values of f:
| f | TWR |
|---|---|
| 0.1 | 1.14 |
| 0.2 | 1.26 |
| 0.25 | 1.31 |
| 0.3 | 1.30 |
| 0.4 | 1.21 |
| 0.5 | 1.00 |
As the table shows, the TWR is maximized at f = 0.25. This is the Optimal f for this particular trade sequence. Any value of f above or below 0.25 will result in a lower terminal wealth. It is also important to note that as f approaches 0.5, the TWR approaches 1.00, and beyond that, it begins to decline, indicating that excessive risk is detrimental to long-term growth.
Conclusion
The mathematical core of Optimal f lies in its focus on maximizing the geometric mean of returns. By understanding the concepts of the Terminal Wealth Relative and the Holding Period Return, traders can appreciate the rigor behind this position-sizing methodology. The iterative process of finding the Optimal f, while computationally intensive, provides a clear and objective framework for determining the ideal level of risk to assume on each trade. This mathematical foundation is what improves Optimal f from a mere heuristic to a sophisticated tool for strategic capital allocation and long-term wealth creation.